The finitistic dimension of rings with Morita context
Let \(T=(T_{ij})\) be a \(2\times 2\) matrix ring, where \(T_{11}\), \(T_{22}\) are rings (associative and unitary), and \(T_{12}\), \(T_{21}\) are bimodules over these rings such that \((R,S,M,N)\) is a Morita context. The author assumes that all left modules are Artinian. Every left \(T\)-module is determined by a quadruple \((P,Q,f,g)\), where \(P\) is a left \(R\)-module, \(Q\) is a left \(S\)-module and \(f\colon M\otimes_SQ\to P\), \(g\colon N\otimes_RP\to Q\) are homomorphisms of modules making commutative certain diagrams. Proposition 3.1 gives a bound for the projective dimension of a left \(T\)-module: \[ \begin{multlined}\text{pd}(P,Q,f,g)\leq 1+\sup\{\text{pd}(_SQ),1+\text{pd}(_SN)+\text{pd}(_{R/(MN)}\ker(f)),\\ \text{pd}(_SN)+\text{pd}(_{R/(MN)}P/f(M\otimes_SQ))\},\end{multlined} \] where \(MN\subseteq R\) denotes one of the trace ideals. The left finitistic dimension \(\text{lfPD}(T)\) of \(T\), defined as the supremum of the projective dimensions of those finitely generated left modules having finite projective dimension, is then studied. For example, the author deduces, in the case that \(M_S\) is flat and \(_SN\) has finite projective dimension, that \(\text{lfPD}(T)\leq\text{lfPD}(S)+\text{LGD}(R/(MN))+2\), where \(\text{LGD}(R/(MN))\) is the left global dimension of \(R/(MN)\).
scientific article; zbMATH DE number 1267192